How can I find the Coefficent of a term when I am multiplying two binomial expansions?

Question

For Example, what is the coeffecint of $x^{-1}$ in the expansion of $$ (\frac{1}{2x}+3x)^5​(x+​1)^4\text{?} $$ How can I find the coefficient without expanding by hand?

Answer 1

$x^{-1}$ can come from $1\cdot x^{-1}$, $x\cdot x^{-2}$, $x^2\cdot x^{-3}$, $x^3\cdot x^{-4}$, or $x^4\cdot x^{-5}$. The coefficient of $x^{-1}$ in the product is the sum of the product of the coefficients of these terms in each binomial. We have $$\sum_{i=0}^4 \frac{1}{32}\begin{pmatrix}4\\i\end{pmatrix}\begin{pmatrix}5\\i+1\end{pmatrix}=\frac{63}{16}$$